A particle $A$ of mass $m$ and initial velocity $v$ collides with a particle $B$ of mass $\frac{m}{2}$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths ${\lambda }_A$ to ${\lambda }_B$ after the collision is:

• A

  $\frac{{\lambda }_A}{{\lambda }_B}=\frac{1}{2}$
• B

  $\frac{{\lambda }_A}{{\lambda }_B}=\frac{1}{3}$
• Correct answer
  C

  $\frac{{\lambda }_A}{{\lambda }_B}=2$
• D

  $\frac{{\lambda }_A}{{\lambda }_B}=\frac{2}{3}$

Solution

$mV=m{V}_A+\frac{m}{2}{V}_B$

$V=V_A+\frac{V_B}{2}------------\text{(i)}$

$V=V_B-V_A------------\text{(ii)}$

$\text{Solving (i) and (ii)}$

$2V=\frac{3V_B}{2}$

$\therefore V_B=\frac{4V}{3}$

$V_A=V_B-V=\frac{V}{3}$

$P_A=\frac{mV}{3}$

$P_B=\frac{m}{2}.\frac{4V}{3}$

$\frac{{\lambda }_A}{{\lambda }_B}=\frac{h\times P_B}{P_A\times h}=\frac{\frac{2mV}{3}}{\frac{mV}{3}}=2$

Hence, the correct answer is option: 'C'.